Manuscript Hawk07-A@1 · Hawk07-A.tex · lemma · f9c0f5da

Lemma: Two-copy twirl

The Haar decoupling calculation · lines 729–760 · proof included or linked

Hawk07-A@1#lem:twirl, Hawk07-A@1#eq:twirl

For $D \geq 2$ ,

E_{U} U^{†\otimes 2} O U^{\otimes 2} = α I + β F,

where

α = \frac{Tr O - Tr ( O F ) / D}{D ^{2} - 1}, β = \frac{Tr ( O F ) - Tr O / D}{D ^{2} - 1} .

For $O = I_{P P^{'}} \otimes F_{B B^{'}}$ ,

α = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1}, β = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1} .

Proof. The averaged operator commutes with every $U \otimes U$ . The symmetric and antisymmetric subspaces give its expression as a linear combination of identity and swap. Preservation of $Tr O$ and $Tr (O F)$ fixes the coefficients. For the specified $O$ , those traces are $r^{2} b$ and $r b^{2}$ .

Displayed formulas in this passage

Hawk07-A@1#eq:twirl · line 746
$α = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1}, β = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1} .$
unlabeled · line 732
$E_{U} U^{†\otimes 2} O U^{\otimes 2} = α I + β F,$
unlabeled · line 738
$α = \frac{Tr O - Tr ( O F ) / D}{D ^{2} - 1}, β = \frac{Tr ( O F ) - Tr O / D}{D ^{2} - 1} .$

Lemma: Two-copy twirl

Displayed formulas in this passage

Nearby in this section